-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathokada85.m
632 lines (550 loc) · 23.2 KB
/
okada85.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
function varargout=okada85(varargin)
%OKADA85 Surface deformation due to a finite rectangular source.
% [uE,uN,uZ,uZE,uZN,uNN,uNE,uEN,uEE] = OKADA85(E,N,...
% DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN,NU)
% computes Okada's 1985 solution for displacements, tilts and strains in a
% geographic referential (East,North,Up). Coordinates (0,0) and depth
% correspond to the fault centroid.
%
% E,N : Coordinates of observation points (relative to fault centroid)
% DEPTH : Depth of the fault centroid (DEP > 0)
% STRIKE : Strike-angle from North (in degrees)
% DIP : Dip-angle (in degrees, must be scalar)
% LENGTH : Fault length in strike direction (LEN > 0)
% WIDTH : Fault width in dip direction (WIDTH > 0)
% RAKE : Slip-angle direction on fault plane (in degrees)
% SLIP : Dislocation in rake direction (length unit)
% OPEN : Dislocation in tensile component (same unit as SLIP)
% NU : Poisson's ratio (optional, default is 0.25)
%
% returns the following variables (same matrix size as E and N):
% uN,uE,uZ : Displacements (unit of SLIP and OPEN)
% uZE,uZN : Tilts (in radian)
% uNN,uNE,uEN,uEE : Strains (unit of SLIP and OPEN)/(Unit of N,E,..,WIDTH)
% POSITIVE = COMPRESSION
%
% Length unit consistency: E, N, DEPTH, LENGTH and WIDTH must have the same unit
% (e.g. km), which can be different from that of SLIP and OPEN (e.g. m).
%
% It is also possible to produce partial outputs, with following syntax:
% [uE,uN,uZ] = OKADA85(...) for displacements only;
% [uE,uN,uZ,uZE,uZN] = OKADA85(...) for displacements and tilts;
% [uE,uN,uZ,uNN,uNE,uEN,uEE] = OKADA85(...) for displacements and strains;
% [uZE,uZN] = OKADA85(...) for tilts only;
% [uZE,uZN,uNN,uNE,uEN,uEE] = OKADA85(...) for tilts and strains;
% [uNN,uNE,uEN,uEE] = OKADA85(...) for strains only.
%
% [...] = OKADA85(...,'plot') or OKADA85(...) without output argument produces
% a 3-D figure with fault geometry and dislocation at scale (if all of the fault
% parameters are scalar).
%
% Example:
% [E,N] = meshgrid(linspace(-10,10,50));
% [uE,uN,uZ] = okada85(E,N,2,30,70,5,3,-45,1,1,'plot');
% figure, surf(E,N,uN)
% considers a 5x3 fault at depth 2, N30°-strike, 70°-dip, and unit dislocation
% in all directions (reverse, senestral and open). Displacements are computed
% on a regular grid from -10 to 10, and North displacements are plotted.
%
% Equations are all vectorized excepted for argument DIP which has to be
% a scalar; all other arguments can be scalar or matrix of the same size.
%
% Formulas and notations similar to Okada [1985], excepted for strain convention
% (here positive strain means compression), and for fault parameters
% after Aki & Richards [1980]:
% STRIKE = fault trace direction (0 to 360° relative to North),
% defined so that the fault dips to the right side of the trace.
% DIP = angle of the fault (0 to 90°, relative to horizontal).
% RAKE = direction the hanging wall moves during rupture, measured
% relative to the fault strike (-180 to 180°).
% Examples:
% DIP=90, RAKE=0 : left lateral (senestral) strike slip
% DIP=90, RAKE=180 : right lateral (dextral) strike slip
% DIP=70, RAKE=90 : reverse fault
% DIP=70, RAKE=-90 : normal fault
%
%
% Author: François Beauducel <[email protected]>
% Institut de Physique du Globe de Paris
% Created: 1997
% Updated: 2011-03-06
%
% References:
% Aki K., and P. G. Richards, Quantitative seismology, Freemann & Co,
% New York, 1980.
% Okada Y., Surface deformation due to shear and tensile faults in a
% half-space, Bull. Seismol. Soc. Am., 75:4, 1135-1154, 1985.
%
% Acknowledgments: Dmitry Nicolsky, University of Alaska
% Development history:
% [2011-03-06]: new optional argument to plot fault geometry with
% output arguments, and bug correction for the fault centroid position
% (in calculation and plot).
% [2010-11-29]: change coordinates and depth to fault centroid
% (instead of middle top edge).
% [2010-09-24]: bugs correction in the syntax of I1, K2 and uyy_tf
% functions, affecting some components. Detected by Dmitry Nicolsky.
%
% Copyright (c) 1997-2010, François Beauducel, covered by BSD License.
% All rights reserved.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are
% met:
%
% * Redistributions of source code must retain the above copyright
% notice, this list of conditions and the following disclaimer.
% * Redistributions in binary form must reproduce the above copyright
% notice, this list of conditions and the following disclaimer in
% the documentation and/or other materials provided with the distribution
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
% AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
% IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
% ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
% LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
% SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
% ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
% POSSIBILITY OF SUCH DAMAGE.
error(nargchk(10,12,nargin))
for i = 1:10
if ~isnumeric(varargin{i})
error('Input arguments E,N,DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN must be numeric.')
end
end
if numel(varargin{5}) ~= 1
error('DIP argument must be scalar.')
end
% Default values for optional input arguments
plotflag = 0; % no plot
nu = 0.25; % isotropic Poisson's ratio
% Assigns input arguments
e = varargin{1};
n = varargin{2};
depth = varargin{3};
strike = varargin{4}*pi/180; % converting STRIKE in radian
dip = varargin{5}*pi/180; % converting DIP in radian ('delta' in Okada's equations)
L = varargin{6};
W = varargin{7};
rake = varargin{8}*pi/180; % converting RAKE in radian
slip = varargin{9};
U3 = varargin{10};
switch nargin
case 11
if isnumeric(varargin{11})
nu = varargin{11};
else
makeplot = varargin{11};
end
case 12
makeplot = varargin{12};
end
if exist('makeplot','var')
if strcmp(makeplot,'plot')
plotflag = 1;
else
error('Unknown last argument.')
end
end
if plotflag & any([numel(depth),numel(strike),numel(L),numel(W),numel(rake),numel(slip),numel(U3)]>1)
warning('Cannot make plot with fault geometry parameters other than scalars.')
plotflag = 0;
end
% Defines dislocation in the fault plane system
U1 = cos(rake).*slip;
U2 = sin(rake).*slip;
% Converts fault coordinates (E,N,DEPTH) relative to centroid
% into Okada's reference system (X,Y,D)
d = depth + sin(dip).*W/2; % d is fault's top edge
ec = e + cos(strike)*cos(dip).*W/2;
nc = n - sin(strike)*cos(dip).*W/2;
x = cos(strike).*nc + sin(strike).*ec + L/2;
y = sin(strike).*nc - cos(strike).*ec + cos(dip).*W;
% Variable substitution (independent from xi and eta)
p = y.*cos(dip) + d.*sin(dip);
q = y.*sin(dip) - d.*cos(dip);
% Displacements
if any(nargout==[3, 5, 7, 9])
ux = -U1/(2*pi) * chinnery(@ux_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@ux_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@ux_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uy = -U1/(2*pi) * chinnery(@uy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uz = -U1/(2*pi) * chinnery(@uz_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uz_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uz_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
ue = sin(strike).*ux - cos(strike).*uy;
un = cos(strike).*ux + sin(strike).*uy;
end
% Tilt
if any(nargout==[2, 5, 6, 9])
uzx = -U1/(2*pi) * chinnery(@uzx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uzx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uzx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uzy = -U1/(2*pi) * chinnery(@uzy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uzy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uzy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
uze = -sin(strike).*uzx + cos(strike).*uzy;
uzn = -cos(strike).*uzx - sin(strike).*uzy;
end
% Strain
if any(nargout==[4, 6, 7, 9])
uxx = -U1/(2*pi) * chinnery(@uxx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uxx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uxx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uxy = -U1/(2*pi) * chinnery(@uxy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uxy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uxy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyx = -U1/(2*pi) * chinnery(@uyx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uyx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uyx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyy = -U1/(2*pi) * chinnery(@uyy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uyy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uyy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
unn = cos(strike)^2*uxx + sin(2*strike)*(uxy + uyx)/2 + sin(strike)^2*uyy;
une = sin(2*strike)*(uxx - uyy)/2 + sin(strike)^2*uyx - cos(strike)^2*uxy;
uen = sin(2*strike)*(uxx - uyy)/2 - cos(strike)^2*uyx + sin(strike)^2*uxy;
uee = sin(strike)^2*uxx - sin(2*strike)*(uyx + uxy)/2 + cos(strike)^2*uyy;
end
% Assigns output arguments
switch nargout
case 2
varargout = {uze, uzn};
case 3
varargout = {ue, un, uz};
case 4
varargout = {unn, une, uen, uee};
case 5
varargout = {ue, un, uz, uze, uzn};
case 6
varargout = {uze, ezn, unn, une, uen, uee};
case 7
varargout = {ue, un, uz, unn, une, uen, uee};
case 9
varargout = {ue, un, uz, uze, uzn, unn, une, uen, uee};
case 0
plotflag = 1;
otherwise
disp('Unvalid number of output arguments.')
end
% no output argument: plots geometry of the fault and dislocation
if plotflag
figure
plot(e,n,'.r','MarkerSize',.1)
alpha = pi/2 - strike;
x_fault = L/2*cos(alpha)*[-1,1,1,-1] + sin(alpha)*cos(dip)*W/2*[-1,-1,1,1];
y_fault = L/2*sin(alpha)*[-1,1,1,-1] + cos(alpha)*cos(dip)*W/2*[1,1,-1,-1];
z_fault = -d + sin(dip)*W*[1,1,0,0];
ddx = U1*cos(alpha) - U2*sin(alpha)*cos(dip) + U3*sin(alpha)*sin(dip);
ddy = U1*sin(alpha) + U2*cos(alpha)*cos(dip) - U3*cos(alpha)*sin(dip);
ddz = U2*sin(dip) + U3*cos(dip);
patch(x_fault,y_fault,z_fault,.3*[1,1,1],'EdgeColor','k','LineWidth',2)
patch(x_fault+ddx/2,y_fault+ddy/2,z_fault+ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
patch(x_fault-ddx/2,y_fault-ddy/2,z_fault-ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
xlabel('East'), ylabel('North'), zlabel('Vertical')
view(3), grid on, axis equal, rotate3d
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Notes for I... and K... subfunctions:
%
% 1. original formulas use Lame's constant as mu/(mu+lambda) which
% depends only on the Poisson's ratio = 1 - 2*nu
% 2. tests for cos(dip) == 0 are made with "cos(dip) > eps"
% because cos(90*pi/180) is not zero but = 6.1232e-17 (!)
% NOTE: don't use cosd and sind because of incompatibility
% with Matlab v6 and earlier...
% =================================================================
% Chinnery's notation [equation (24) p. 1143]
% -----------------------------------------------------------------
function u=chinnery(f,x,p,L,W,q,dip,nu)
u = feval(f,x,p,q,dip,nu) ...
- feval(f,x,p-W,q,dip,nu) ...
- feval(f,x-L,p,q,dip,nu) ...
+ feval(f,x-L,p-W,q,dip,nu);
% =================================================================
% Displacement subfunctions
% strike-slip displacement subfunctions [equation (25) p. 1144]
% -----------------------------------------------------------------
function u=ux_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./(R.*(R + eta)) ...
+ atan(xi.*eta./(q.*R)) ...
+ I1(xi,eta,q,dip,nu,R).*sin(dip);
% -----------------------------------------------------------------
function u=uy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + eta)) ...
+ q.*cos(dip)./(R + eta) ...
+ I2(eta,q,dip,nu,R).*sin(dip);
% -----------------------------------------------------------------
function u=uz_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = (eta*sin(dip) - q*cos(dip)).*q./(R.*(R + eta)) ...
+ q.*sin(dip)./(R + eta) ...
+ I4(db,eta,q,dip,nu,R).*sin(dip);
% dip-slip displacement subfunctions [equation (26) p. 1144]
% -----------------------------------------------------------------
function u=ux_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q./R ...
- I3(eta,q,dip,nu,R).*sin(dip).*cos(dip);
% -----------------------------------------------------------------
function u=uy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + xi)) ...
+ cos(dip).*atan(xi.*eta./(q.*R)) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);
% -----------------------------------------------------------------
function u=uz_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = db.*q./(R.*(R + xi)) ...
+ sin(dip).*atan(xi.*eta./(q.*R)) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).*cos(dip);
% tensile fault displacement subfunctions [equation (27) p. 1144]
% -----------------------------------------------------------------
function u=ux_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2 ./(R.*(R + eta)) ...
- I3(eta,q,dip,nu,R).*sin(dip).^2;
% -----------------------------------------------------------------
function u=uy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -(eta*sin(dip) - q*cos(dip)).*q./(R.*(R + xi)) ...
- sin(dip).*(xi.*q./(R.*(R + eta)) ...
- atan(xi.*eta./(q.*R))) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).^2;
% -----------------------------------------------------------------
function u=uz_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + xi)) ...
+ cos(dip).*(xi.*q./(R.*(R + eta)) ...
- atan(xi.*eta./(q.*R))) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).^2;
% I... displacement subfunctions [equations (28) (29) p. 1144-1145]
% -----------------------------------------------------------------
function I=I1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (-xi./(cos(dip).*(R + db))) - sin(dip)./cos(dip) .*I5(xi,eta,q,dip,nu,R,db);
else
I = -(1 - 2*nu)/2 * xi.*q./(R + db).^2;
end
% -----------------------------------------------------------------
function I=I2(eta,q,dip,nu,R)
I = (1 - 2*nu) * (-log(R + eta)) - I3(eta,q,dip,nu,R);
% -----------------------------------------------------------------
function I=I3(eta,q,dip,nu,R)
yb = eta*cos(dip) + q*sin(dip);
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (yb./(cos(dip)*(R + db)) - log(R + eta)) ...
+ sin(dip)./cos(dip) * I4(db,eta,q,dip,nu,R);
else
I = (1 - 2*nu)/2 * (eta./(R + db) + yb.*q./(R + db).^2 - log(R + eta));
end
% -----------------------------------------------------------------
function I=I4(db,eta,q,dip,nu,R)
if cos(dip) > eps
I = (1 - 2*nu) * 1/cos(dip) * (log(R + db) - sin(dip)*log(R + eta));
else
I = -(1 - 2*nu) * q./(R + db);
end
% -----------------------------------------------------------------
function I=I5(xi,eta,q,dip,nu,R,db)
X = sqrt(xi.^2 + q.^2);
if cos(dip) > eps
I = (1 - 2*nu) * 2./cos(dip) ...
.* atan((eta.*(X + q.*cos(dip)) + X.*(R + X).*sin(dip))./(xi.*(R + X).*cos(dip)));
else
I = -(1 - 2*nu) * xi.*sin(dip)./(R + db);
end
% =================================================================
% Tilt subfunctions
% strike-slip tilt subfunctions [equation (37) p. 1147]
% -----------------------------------------------------------------
function u=uzx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -xi.*q.^2.*A(eta,R).*cos(dip) ...
+ ((xi.*q)./R.^3 - K1(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uzy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (db.*q./R.^3).*cos(dip) ...
+ (xi.^2.*q.*A(eta,R)*cos(dip) - sin(dip)./R + yb.*q./R.^3 - K2(xi,eta,q,dip,nu,R))*sin(dip);
% dip-slip tilt subfunctions [equation (38) p. 1147]
% -----------------------------------------------------------------
function u=uzx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = db.*q./R.^3 ...
+ q*sin(dip)./(R.*(R + eta)) ...
+ K3(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uzy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*db.*q.*A(xi,R) ...
- (2*db./(R.*(R + xi)) + xi*sin(dip)./(R.*(R + eta)))*sin(dip) ...
+ K1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% tensile fault tilt subfunctions [equation (39) p. 1147]
% -----------------------------------------------------------------
function u=uzx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3*sin(dip) ...
- q.^3.*A(eta,R)*cos(dip) ...
+ K3(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uzy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (yb*sin(dip) + db*cos(dip)).*q.^2.*A(xi,R) ...
+ xi.*q.^2.*A(eta,R)*sin(dip)*cos(dip) ...
- (2*q./(R.*(R + xi)) - K1(xi,eta,q,dip,nu,R))*sin(dip)^2;
% -----------------------------------------------------------------
function a=A(x,R)
a = (2*R + x)./(R.^3.*(R + x).^2);
% K... tilt subfunctions [equations (40) (41) p. 1148]
% -----------------------------------------------------------------
function K=K1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
K = (1 - 2*nu) * xi/cos(dip) .* (1./(R.*(R + db)) - sin(dip)./(R.*(R + eta)));
else
K = (1 - 2*nu) * xi.*q./(R + db).^2;
end
% -----------------------------------------------------------------
function K=K2(xi,eta,q,dip,nu,R)
K = (1 - 2*nu) * (-sin(dip)./R + q*cos(dip)./(R.*(R + eta))) - K3(xi,eta,q,dip,nu,R);
% -----------------------------------------------------------------
function K=K3(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
if cos(dip) > eps
K = (1 - 2*nu) * 1/cos(dip) .* (q./(R.*(R + eta)) - yb./(R.*(R + db)));
else
K = (1 - 2*nu) * sin(dip)./(R + db) .* (xi.^2./(R.*(R + db)) - 1);
end
% =================================================================
% Strain subfunctions
% strike-slip strain subfunctions [equation (31) p. 1145]
% -----------------------------------------------------------------
function u=uxx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.^2.*q.*A(eta,R) ...
- J1(xi,eta,q,dip,nu,R)*sin(dip);
% -----------------------------------------------------------------
function u=uxy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = xi.^3.*db./(R.^3.*(eta.^2 + q.^2)) ...
- (xi.^3.*A(eta,R) + J2(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uyx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3*cos(dip) ...
+ (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uyy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3*cos(dip) ...
+ (q.^3.*A(eta,R)*sin(dip) - 2*q*sin(dip)./(R.*(R + eta)) ...
- (xi.^2 + eta.^2)./R.^3*cos(dip) - J4(xi,eta,q,dip,nu,R))*sin(dip);
% dip-slip strain subfunctions [equation (32) p. 1146]
% -----------------------------------------------------------------
function u=uxx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3 ...
+ J3(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uxy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3 ...
- sin(dip)./R ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uyx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3 ...
+ q*cos(dip)./(R.*(R + eta)) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uyy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.^2.*q.*A(xi,R) ...
- (2*yb./(R.*(R + xi)) + xi*cos(dip)./(R.*(R + eta)))*sin(dip) ...
+ J2(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% tensile fault strain subfunctions [equation (33) p. 1146]
% -----------------------------------------------------------------
function u=uxx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q.^2.*A(eta,R) ...
+ J3(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uxy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = -db.*q./R.^3 ...
- xi.^2.*q.*A(eta,R)*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uyx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3*cos(dip) ...
+ q.^3.*A(eta,R)*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uyy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (yb*cos(dip) - db*sin(dip)).*q.^2.*A(xi,R) ...
- q*sin(2*dip)./(R.*(R + xi)) ...
- (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R))*sin(dip)^2;
% J... tensile fault subfunctions [equations (34) (35) p. 1146-1147]
% -----------------------------------------------------------------
function J=J1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1/cos(dip) * (xi.^2./(R.*(R + db).^2) - 1./(R + db)) ...
- sin(dip)/cos(dip)*K3(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * q./(R + db).^2 .* (2*xi.^2./(R.*(R + db)) - 1);
end
% -----------------------------------------------------------------
function J=J2(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1/cos(dip) * xi.*yb./(R.*(R + db).^2) ...
- sin(dip)/cos(dip)*K1(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * xi*sin(dip)./(R + db).^2 .* (2*q.^2./(R.*(R + db)) - 1);
end
% -----------------------------------------------------------------
function J=J3(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * -xi./(R.*(R + eta)) ...
- J2(xi,eta,q,dip,nu,R);
% -----------------------------------------------------------------
function J=J4(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * (-cos(dip)./R - q*sin(dip)./(R.*(R + eta))) ...
- J1(xi,eta,q,dip,nu,R);